Electric-field-induced Majorana Fermions in Armchair Carbon Nanotubes

We consider theoretically an armchair carbon nanotube (CNT) in the presence of an electric field and in contact with an $s$-wave superconductor. We show that the proximity effect opens up superconducting gaps in the CNT of different strengths for the exterior and interior branches of the two Dirac points. For strong proximity induced superconductivity the interior gap can be of the $p$-wave type, while the exterior gap can be tuned by the electric field to be of the $s$-wave type. Such a setup supports a single Majorana bound state at each end of the CNT. In the case of a weak proximity induced superconductivity, the gaps in both branches are of the $p$-wave type. However, the temperature can be chosen in such a way that the smallest gap is effectively closed. Using renormalization group techniques we show that the Majorana bound states exist even after taking into account electron-electron interactions.

Figure 1

(a) An armchair nanotube (cylinder) is placed on top of a superconductor (blue slab). The $x$-axis points along the nanotube. An electric field $E$ is applied perpendicular to the nanotube, say along $y$-axis [19]. There are two nonequivalent lattice sites: $A$ (light red) and $B$ (light green). (b) The distances between the superconductor surface and the atoms of sublattice $A$ (dark-red row) and of sublattice $B$ (dark-green row) are assumed the same. Thus, the tunneling amplitudes to the different sublattices are (nearly) equal.

Figure 2

The energy spectrum around the Dirac points $K$, $K^{\prime }$ for a (10,10)-CNT in an electric field $E=1\mathrm{V}/\mathrm{nm}$, which consists of exterior (full line) and interior (dashed line) branches. Each branch of the spectrum is characterized by the sign of the spin projection along the $z$-axis $⟨S_z⟩$ (red: spin down, blue: spin up). The Fermi level $\mu$ lies inside the gap given by $2eE\xi$, and $\delta =eE\xi +\alpha -\mu$.

Figure 3

The particle-hole spectrum of a CNT (10,10) in the presence of an electric field $E$ with the Fermi level $\mu$ tuned inside the energy gap between the two upper branches (solid black lines). All energies are counted from $\mu =0.11\mathrm{meV}$ (see Fig. 2). By proximity effect, superconducting gaps ${\Delta }_{e,i}$ are opened at the Fermi points $k_F$. (a) Here, the $E$-field is fixed at $1\mathrm{V}/\mathrm{nm}$ and ${\Delta }_d$ is varied. For ${\Delta }_d=5\mu \mathrm{eV}<{\Delta }_{c1}$ both branches are in the $p$-wave phase (dotted blue line). At the critical value ${\Delta }_d=23\mu \mathrm{eV}={\Delta }_{c1}$, the gaps induced by the proximity effect and by $E$ are equal (dot-dashed red line). For ${\Delta }_d=30\mu \mathrm{eV}>{\Delta }_{c1}$ only the interior branch is in the $p$-wave phase (dashed green line). Keeping ${\Delta }_d$ constant at $11\mu \mathrm{eV}$ and changing $E$, one goes from a regime [dashed blue line in (b)] with $E=0.4\mathrm{V}/\mathrm{nm}<{\mathrm{E}}_{c1}=0.6\mathrm{V}/\mathrm{nm}$ where only the interior branch is in the $p$-wave phase to a regime [dashed green line (c)] with $E=1\mathrm{V}/\mathrm{nm}>E_{c1}=0.6\mathrm{V}/\mathrm{nm}$ where both branches are in the $p$-wave phase.